3.102 \(\int \frac{(a+b x^2)^8}{x^{21}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b \left (a+b x^2\right )^9}{180 a^2 x^{18}}-\frac{\left (a+b x^2\right )^9}{20 a x^{20}} \]

[Out]

-(a + b*x^2)^9/(20*a*x^20) + (b*(a + b*x^2)^9)/(180*a^2*x^18)

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Rubi [A]  time = 0.0170586, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{b \left (a+b x^2\right )^9}{180 a^2 x^{18}}-\frac{\left (a+b x^2\right )^9}{20 a x^{20}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^21,x]

[Out]

-(a + b*x^2)^9/(20*a*x^20) + (b*(a + b*x^2)^9)/(180*a^2*x^18)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^8}{x^{21}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^9}{20 a x^{20}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^2\right )}{20 a}\\ &=-\frac{\left (a+b x^2\right )^9}{20 a x^{20}}+\frac{b \left (a+b x^2\right )^9}{180 a^2 x^{18}}\\ \end{align*}

Mathematica [B]  time = 0.0042056, size = 106, normalized size = 2.65 \[ -\frac{7 a^6 b^2}{4 x^{16}}-\frac{4 a^5 b^3}{x^{14}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{28 a^3 b^5}{5 x^{10}}-\frac{7 a^2 b^6}{2 x^8}-\frac{4 a^7 b}{9 x^{18}}-\frac{a^8}{20 x^{20}}-\frac{4 a b^7}{3 x^6}-\frac{b^8}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^21,x]

[Out]

-a^8/(20*x^20) - (4*a^7*b)/(9*x^18) - (7*a^6*b^2)/(4*x^16) - (4*a^5*b^3)/x^14 - (35*a^4*b^4)/(6*x^12) - (28*a^
3*b^5)/(5*x^10) - (7*a^2*b^6)/(2*x^8) - (4*a*b^7)/(3*x^6) - b^8/(4*x^4)

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Maple [B]  time = 0.005, size = 91, normalized size = 2.3 \begin{align*} -{\frac{7\,{a}^{2}{b}^{6}}{2\,{x}^{8}}}-{\frac{28\,{a}^{3}{b}^{5}}{5\,{x}^{10}}}-{\frac{7\,{a}^{6}{b}^{2}}{4\,{x}^{16}}}-{\frac{35\,{a}^{4}{b}^{4}}{6\,{x}^{12}}}-{\frac{{b}^{8}}{4\,{x}^{4}}}-{\frac{4\,a{b}^{7}}{3\,{x}^{6}}}-4\,{\frac{{a}^{5}{b}^{3}}{{x}^{14}}}-{\frac{{a}^{8}}{20\,{x}^{20}}}-{\frac{4\,{a}^{7}b}{9\,{x}^{18}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^21,x)

[Out]

-7/2*a^2*b^6/x^8-28/5*a^3*b^5/x^10-7/4*a^6*b^2/x^16-35/6*a^4*b^4/x^12-1/4*b^8/x^4-4/3*a*b^7/x^6-4*a^5*b^3/x^14
-1/20*a^8/x^20-4/9*a^7*b/x^18

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Maxima [B]  time = 2.65241, size = 124, normalized size = 3.1 \begin{align*} -\frac{45 \, b^{8} x^{16} + 240 \, a b^{7} x^{14} + 630 \, a^{2} b^{6} x^{12} + 1008 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 720 \, a^{5} b^{3} x^{6} + 315 \, a^{6} b^{2} x^{4} + 80 \, a^{7} b x^{2} + 9 \, a^{8}}{180 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^21,x, algorithm="maxima")

[Out]

-1/180*(45*b^8*x^16 + 240*a*b^7*x^14 + 630*a^2*b^6*x^12 + 1008*a^3*b^5*x^10 + 1050*a^4*b^4*x^8 + 720*a^5*b^3*x
^6 + 315*a^6*b^2*x^4 + 80*a^7*b*x^2 + 9*a^8)/x^20

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Fricas [B]  time = 1.27565, size = 217, normalized size = 5.42 \begin{align*} -\frac{45 \, b^{8} x^{16} + 240 \, a b^{7} x^{14} + 630 \, a^{2} b^{6} x^{12} + 1008 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 720 \, a^{5} b^{3} x^{6} + 315 \, a^{6} b^{2} x^{4} + 80 \, a^{7} b x^{2} + 9 \, a^{8}}{180 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^21,x, algorithm="fricas")

[Out]

-1/180*(45*b^8*x^16 + 240*a*b^7*x^14 + 630*a^2*b^6*x^12 + 1008*a^3*b^5*x^10 + 1050*a^4*b^4*x^8 + 720*a^5*b^3*x
^6 + 315*a^6*b^2*x^4 + 80*a^7*b*x^2 + 9*a^8)/x^20

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Sympy [B]  time = 1.17487, size = 99, normalized size = 2.48 \begin{align*} - \frac{9 a^{8} + 80 a^{7} b x^{2} + 315 a^{6} b^{2} x^{4} + 720 a^{5} b^{3} x^{6} + 1050 a^{4} b^{4} x^{8} + 1008 a^{3} b^{5} x^{10} + 630 a^{2} b^{6} x^{12} + 240 a b^{7} x^{14} + 45 b^{8} x^{16}}{180 x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**21,x)

[Out]

-(9*a**8 + 80*a**7*b*x**2 + 315*a**6*b**2*x**4 + 720*a**5*b**3*x**6 + 1050*a**4*b**4*x**8 + 1008*a**3*b**5*x**
10 + 630*a**2*b**6*x**12 + 240*a*b**7*x**14 + 45*b**8*x**16)/(180*x**20)

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Giac [B]  time = 2.13553, size = 124, normalized size = 3.1 \begin{align*} -\frac{45 \, b^{8} x^{16} + 240 \, a b^{7} x^{14} + 630 \, a^{2} b^{6} x^{12} + 1008 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 720 \, a^{5} b^{3} x^{6} + 315 \, a^{6} b^{2} x^{4} + 80 \, a^{7} b x^{2} + 9 \, a^{8}}{180 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^21,x, algorithm="giac")

[Out]

-1/180*(45*b^8*x^16 + 240*a*b^7*x^14 + 630*a^2*b^6*x^12 + 1008*a^3*b^5*x^10 + 1050*a^4*b^4*x^8 + 720*a^5*b^3*x
^6 + 315*a^6*b^2*x^4 + 80*a^7*b*x^2 + 9*a^8)/x^20